Optimization in Brachytherapy. Gary A. Ezzell, Ph.D. Mayo Clinic Scottsdale

Transcription

1 Optimization in Brachytherapy Gary A. Ezzell, Ph.D. Mayo Clinic Scottsdale

2 Outline General concepts of optimization Classes of optimization techniques Concepts underlying some commonly available methods Specific brachytherapy applications - Permanent prostate implants - High dose rate brachytherapy

3 General concepts of optimization Big subject with huge literature This treatment will not be particularly mathematical Focus on key concepts and how they apply to brachytherapy The concepts also apply to IMRT optimization

4 Examples of brachytherapy optimization Permanent prostate implants - Fixed seed activity - Choose seed locations to meet some objectives target coverage dose uniformity rectal/urethral sparing

5 Examples of brachytherapy optimization HDR brachytherapy - Fixed activity, variable dwell times - Choose dwell positions and times - e.g. two-catheter endobronchial implant: sufficiently uniform dose at 1 cm from each of the catheters

6 Generalized brachytherapy problem Design a distribution of source terms such that the resultant dose distribution satisfies certain constraints and meets certain objectives as well as possible

7 Free variables Source locations, source strengths, and/or dwell times Depends on the specific problem - Prostate seed implant with template - HDR with implanted applicators - Stereotactic brain implants

8 Constraints and Feasibilty Hard constraints: cannot be violated - Physical (non-negative sources) - Clinical (cord dose < 45 Gy) Soft constraints: violation reduces plan quality (e.g. Rx dose to 98% of PTV) Any plan that satisfies the constraints is feasible

9 Feasible vs. Optimal Sometimes, a feasible solution is clinically acceptable More interesting problem: find a solution that optimizes some objective - e.g. the dose to the surface of the prostate PTV is to match the prescription dose as closely as possible.

10 Express as a minimization problem Minimize the variance of the doses D i at points i on the PTV surface from the prescription dose D p Minimize f = (D i D p ) 2 f is a simple objective function

11 Two competing objectives Prostate example: - Minimize the dose variation on the surface of the PTV - Minimize the dose to the adjacent rectum Cannot do both, must balance the two goals

12 Dealing with multiple objectives Most common approach: combine into a single objective function with weighting factors to control their relative influence Minimize f = w 1 (D i D p ) 2 + w 2 (D j L r, if >0, 0 otherwise) Uniformity on surface of PTV Rectal doses below limit L r

13 Dealing with multiple objectives Minimize f = w 1 (D i D p ) 2 + w 2 (D j L r, if >0, 0 otherwise)

14 Dealing with multiple objectives Minimize f = w 1 (D i D p ) 2 + w 2 (D j L r, if >0, 0 otherwise)

15 Additional objectives Add penalty based on number of needles Minimize f = w 1 (D i D p ) 2 + w 2 (D j L r, if >0, 0 otherwise) + w 3 N

16 How good is that function? The objective function is a mathematical model of the clinical goals. Does the model capture the essence of the clinical thinking? Minimize f = w 1 (D i D p ) 2 + w 2 (D j L r, if >0, 0 otherwise)?

17 How to steer the results? What are the optimization parameters? Minimize f = w 1 (D i D p ) 2 + w 2 (D j L r, if >0, 0 otherwise)

18 How to steer the results? Aren t these chosen by the physician? Minimize f = w 1 (D i D p ) 2 + w 2 (D j L r, if >0, 0 otherwise)

19 How to steer the results? Aren t these chosen by the physician? Minimize f = w 1 (D i D p ) 2 + w 2 (D j L r, if >0, 0 otherwise) Yes and no. - It may be that the doses the physician really wants are not the best to use as input

20 Three uses of the word prescription Prescription - The dose objectives you want to achieve - The dose parameters you give to the optimizer - The dose results you eventually accept Ideally, all are the same. Not always the case

21 An idealized problem D 0 = A + B D 1 = D 5 = A + B/5 D 2 = D 4 = A/5 + B D 3 = A/9 + B D 6 = A + B/9 D R = A/2 + B/2 Dose depends only on distance, no attenuation 1 source strength unit at unit distance 1 dose unit

22 Target is to receive at least 5 dose units

23 Rectum is not to exceed 5 units of dose

24 Feasible solutions for both constraints

25 Optimize dose to rectum Optimum: A=B=4.5

26 Features of linear systems Feasible solutions to sets of linear constraints are bounded by a convex multidimensional polyhedron. (This simple example is two-dimensional.) Convexity means that you can move from one feasible solution to another along a straight vector in the space without ever leaving the feasible region.

27 Features of linear systems Inequalities define regions of the space. Equalities define surfaces of lower dimensionality. (DR<5 defines a section of a plane, while DR=5 defines a line.)

28 Features of linear systems When an optimum solution is sought by maximizing or minimizing a linear function of the variables, the optimal solution will lie on a vertex.

29 Quadratic functions For quadratic functions, the surfaces are curved, but similar concepts apply

30 More interesting problem Move rectum closer; point R now at 0.8 units from origin

31 Competing objectives T 6 = ( Di i= 1 5) 2 / 6 Average deviation of target dose points from prescription 5 S = D R = A/ B/1.64 Rectal dose point Combine in an objective function with normalized weighting factors F = (w 1 T + w 2 S) / (w 1 + w 2 )

32 Variation of Target component

33 Variation of Structure component

34 Solution space for w 2 =100, w 1 =1

35 Solution space for w 2 =5, w 1 =1

36 Solution space for w 2 =3, w 1 =1

37 Location of optimum shifts with weights More uniform More sparing

38 Optimization tradeoff: Pareto front More uniform Feasible, not optimal More sparing

39 Optimality: no objective can be improved without worsening another one Can improve T with S constant Can improve S with T constant Cannot improve S without violating T

40 Practical example of Pareto front If a physician wants to cover a prostate while sparing the rectum, the planner can provide a series of plans, with, for example, 98%, 95%, 93%, and 90% target coverage, each with the best obtainable rectal dose. Such a series of results represents the Pareto front for the problem at hand and shows the best available choices.

41 General approach to optimization Decide which objective is most important Decide the limit, or range of limits, to which that objective may be pushed Optimize the other objectives by pushing the primary objective to this limit

42 Related issue: the evaluation problem The objective function is not available to the user, only the steering parameters The function may not encode the quality measure preferred by the planner: e.g. DVH, conformity index, TCP, EUD,

43 Evaluation problem Carefully think through how to quantitatively evaluate the results Use the mechanics of the optimizer software to control those results Cannot assume that what the computer declares is mathematically optimal according to its algorithm is actually clinically optimal

44 Robustness problem When planning an implant, need to think about the effect of uncertainty in carrying it out e.g. for prostate: A plan with more seeds of somewhat lower activity may be more stable in the face of uncertainty than one with fewer seeds of higher activity

45 Local vs global optimum Some objective functions can have multiple minima (e.g. with dose/volume constraints; multiple possible source positions) Some search process can escape from local minima, some cannot (more later)

46 Classes of optimization techniques Most approaches are iterative

47 Classes of optimization techniques Most approaches are iterative Differ in how to create new solutions

48 Classes of optimization techniques Deterministic Stochastic (probabilistic) Deductive Heuristic, or phenomenological

49 Deterministic methods Move downhill from the starting point according to an algorithm (steepest descent, conjugate gradient, Nelder- Mead simplex, ) May use calculations of gradient in solution space

50 Deterministic methods Fast, but cannot escape from local minima (may not be a problem!)

51 Stochastic (probabilistic) methods Use randomness in the search process (simulated annealing, genetic algorithms) Can be applied to objective functions of any mathematical form

52 Stochastic methods Slower (but may not be a problem!) Can escape from local minima

53 Deductive methods Test solutions in a systematic, logical sequence that is designed to eliminate subsets of potential solutions from consideration (e.g. branch and bound) Can be applied to objective functions of any mathematical form

54 Heuristic methods Use related relationships to arrive at an adequate solution to the original problem (e.g. geometric optimization) Related relationships sometimes are called adjoint functions Can be very fast and produce results that are useful without being optimal

55 More on two probabilistic methods

56 Simulated annealing Analogous to a crystal seeking its lowest energy state as it slowly cools from an initially high temperature (Metropolis 1953) Early application to brachytherapy by Sloboda (1992) At least one current commercial application: VariSeed

57 Simulated annealing algorithm 1. Choose initial solution and evaluate 2. Create another in by taking a step in solution space in a random direction The size of the step depends on a temperature parameter, initially large. The temperature is reduced as the iteration proceeds

58 Simulated annealing algorithm 3. Evaluate the new solution and compare to previous (ΔF). Accept if better. If worse, still accept with a probability that depends on the temperature and the size of the deterioration p = e ΔF / kt

59 Simulated annealing algorithm 4. Iterate, using a cooling schedule, thus shortening the steps and reducing the likelihood of uphill moves. 5. By reducing the temperature sufficiently slowly, the system is allowed find the region of solution space that contains the global optimum and converge to it

60 Simulated annealing algorithm Slow (logarithmic) cooling can guarantee global optimality Faster cooling schedules and different probability distribution forms speed the convergence

61 Genetic algorithms Operate on populations of alternative solutions Mimic processes of evolution - Mutation - Mating with crossover of genes - Replication in the next generation proportional to fitness

62 Genetic algorithms 1. Construct a population of individual solutions, each being a string of numbers encoding the values of the free variables. Typically, each solution is a bit string of concatenated binary numbers e.g. for prostate implant: 1 means seed position is on ; 0 means is off

63 Genetic algorithms 2. Evaluate each individual using the objective function, and then assigning a fitness value to each 3. Produce a new population, or generation, from the old using a combination of operators

64 Genetic operators a. Proportional replication: select a string for representation in the new generation with a probability dependent on its fitness b. Crossover: pairs of selected strings mate and produce offspring that contain parts of each parent c. Mutation: introduce variation by altering randomly selected bits in some of the strings

65 Crossover Profitable patterns are perpetuated

66 Best solutions mate most frequently quarterback cheerleader medical physicist Profitable patterns are improved

67 Genetic algorithms: population converges to good solutions Yu and Reinstein (1996) described a genetic algorithm that was later developed into clinical tool, Prostate Implant Planning Engine for Radiotherapy, PIPER, (Yu et al. 1999) and used for intraoperative planning (Messing et al. 1999).

68 Geometric Optimization Heuristic method developed by Edmundson (1993) for HDR dwell times Assume that sources are distributed throughout the implant volume (prostate, breast, ) Want dose uniformity between the sources Do not want to define calculation points between the sources

69 Geometric Optimization Look at source locations themselves Want the dose to each location from all the other sources to be uniform Set dwell time for each to be inversely proportional to the sum of the inverse square of the distances to the other sources 1 T i Sources 1 2 = 1, j j i rij

70 Commercial systems Nucletron - Dose point optimization (analytical solution by singular value decomposition) - Geometric optimization BrachyVision - Dose optimization (downhill search by Nelder- Mead simplex) - Geometric optimization VariSeed - Simulated annealing

71 Courtesy Michael Mariscal, Varian Intraoperative planning

72 Dose optimization Courtesy Michael Mariscal, Varian

73 Dose optimization Geometric optimization Courtesy Michael Mariscal, Varian

74 Two-plane breast template (Nucletron) From 1994 brachytherapy summer school

75 Not optimized Optimized Rx line Rx line Unoptimized implants must extend beyond target, optimized do not From 1994 brachytherapy summer school

76 Not optimized Optimized Optimized implants may have locally high doses at the implant boundaries; does this make clinical sense? Dose uniformity in brachytherapy never exists and is its chief advantage; think carefully about what you want; maybe differential dosing is preferable From 1994 brachytherapy summer school

77 Brachytherapy generalization Inverse square is king. - Dwell time optimization is more effective at reducing hot spots than fixing cold spots - Plans with more sources are more robust and forgiving of placement errors than plans with fewer

78 Summary Optimization in brachytherapy always begins with a numerical model of the clinical problem - Model of the implant itself and the biological structures it resides in and near - Model of what is to be accomplished; this is encoded in the objective function The optimizing algorithm searches through potential solutions to find one that best minimizes (or maximizes) the objective function

79 If the modeling has been done well, then that solution will be a good clinical solution, with luck the best available However, since the models are always imperfect and approximate, the results need to be evaluated according to criteria both objective and subjective

80 There is seldom a single, best answer There usually are multiple objectives that compete with each other The planner uses whatever tools there are to steer the optimizer along the Pareto front, developing a set of possible solutions that cannot be improved upon in any one area without losing quality in another Our job is to define those choices, and the clinician s is to choose between them

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